Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 4 times and lost 6 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Car
6. Car
7. Goat
8. Goat
9. Goat
10. Car

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 312 times and lost 688 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat
11. Car
12. Car
13. Goat
14. Car
15. Goat
16. Goat
17. Car
18. Car
19. Goat
20. Goat
21. Goat
22. Goat
23. Goat
24. Car
25. Goat
26. Goat
27. Goat
28. Car
29. Car
30. Goat
31. Goat
32. Car
33. Goat
34. Goat
35. Goat
36. Goat
37. Car
38. Goat
39. Goat
40. Car
41. Car
42. Goat
43. Car
44. Car
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Car
52. Goat
53. Goat
54. Car
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Goat
61. Goat
62. Car
63. Car
64. Car
65. Goat
66. Goat
67. Goat
68. Goat
69. Goat
70. Goat
71. Goat
72. Goat
73. Goat
74. Goat
75. Car
76. Car
77. Car
78. Goat
79. Goat
80. Goat
81. Car
82. Goat
83. Goat
84. Goat
85. Car
86. Goat
87. Goat
88. Goat
89. Car
90. Car
91. Goat
92. Car
93. Car
94. Car
95. Goat
96. Goat
97. Goat
98. Goat
99. Car
100. Car
101. Goat
102. Car
103. Goat
104. Goat
105. Goat
106. Goat
107. Car
108. Goat
109. Goat
110. Car
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Goat
117. Car
118. Goat
119. Goat
120. Goat
121. Goat
122. Goat
123. Goat
124. Goat
125. Goat
126. Car
127. Car
128. Goat
129. Goat
130. Goat
131. Goat
132. Car
133. Goat
134. Goat
135. Goat
136. Goat
137. Goat
138. Goat
139. Goat
140. Goat
141. Goat
142. Car
143. Goat
144. Goat
145. Car
146. Goat
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Goat
154. Car
155. Goat
156. Car
157. Goat
158. Car
159. Car
160. Goat
161. Car
162. Car
163. Goat
164. Goat
165. Goat
166. Goat
167. Goat
168. Goat
169. Car
170. Car
171. Goat
172. Goat
173. Goat
174. Goat
175. Goat
176. Car
177. Goat
178. Goat
179. Car
180. Goat
181. Goat
182. Goat
183. Goat
184. Car
185. Car
186. Car
187. Goat
188. Goat
189. Goat
190. Goat
191. Car
192. Car
193. Goat
194. Car
195. Goat
196. Car
197. Goat
198. Goat
199. Goat
200. Goat
201. Goat
202. Car
203. Car
204. Goat
205. Goat
206. Goat
207. Goat
208. Goat
209. Goat
210. Goat
211. Goat
212. Car
213. Car
214. Goat
215. Goat
216. Goat
217. Car
218. Car
219. Goat
220. Goat
221. Goat
222. Car
223. Car
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Goat
230. Goat
231. Goat
232. Goat
233. Car
234. Goat
235. Goat
236. Goat
237. Car
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Car
244. Car
245. Goat
246. Goat
247. Goat
248. Goat
249. Goat
250. Goat
251. Goat
252. Car
253. Car
254. Goat
255. Goat
256. Goat
257. Goat
258. Goat
259. Car
260. Car
261. Goat
262. Goat
263. Car
264. Car
265. Goat
266. Goat
267. Car
268. Car
269. Car
270. Goat
271. Car
272. Goat
273. Goat
274. Car
275. Goat
276. Goat
277. Goat
278. Goat
279. Goat
280. Goat
281. Goat
282. Car
283. Car
284. Goat
285. Goat
286. Goat
287. Goat
288. Goat
289. Car
290. Goat
291. Goat
292. Goat
293. Goat
294. Goat
295. Goat
296. Car
297. Car
298. Goat
299. Goat
300. Goat
301. Car
302. Goat
303. Goat
304. Car
305. Goat
306. Goat
307. Goat
308. Goat
309. Goat
310. Goat
311. Goat
312. Goat
313. Goat
314. Goat
315. Car
316. Car
317. Goat
318. Goat
319. Goat
320. Goat
321. Goat
322. Car
323. Car
324. Car
325. Car
326. Car
327. Goat
328. Goat
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Car
335. Goat
336. Car
337. Goat
338. Car
339. Goat
340. Goat
341. Goat
342. Goat
343. Goat
344. Goat
345. Goat
346. Car
347. Goat
348. Goat
349. Goat
350. Goat
351. Car
352. Goat
353. Goat
354. Car
355. Car
356. Goat
357. Goat
358. Car
359. Car
360. Goat
361. Goat
362. Car
363. Goat
364. Goat
365. Goat
366. Goat
367. Car
368. Goat
369. Goat
370. Car
371. Goat
372. Goat
373. Car
374. Goat
375. Car
376. Goat
377. Goat
378. Goat
379. Goat
380. Goat
381. Car
382. Goat
383. Goat
384. Goat
385. Goat
386. Car
387. Goat
388. Goat
389. Car
390. Car
391. Goat
392. Goat
393. Goat
394. Car
395. Goat
396. Goat
397. Goat
398. Goat
399. Goat
400. Car
401. Goat
402. Car
403. Car
404. Goat
405. Goat
406. Goat
407. Goat
408. Goat
409. Car
410. Car
411. Goat
412. Car
413. Goat
414. Goat
415. Car
416. Car
417. Goat
418. Car
419. Goat
420. Goat
421. Goat
422. Car
423. Goat
424. Goat
425. Goat
426. Goat
427. Goat
428. Goat
429. Goat
430. Car
431. Goat
432. Car
433. Car
434. Car
435. Car
436. Goat
437. Goat
438. Goat
439. Goat
440. Goat
441. Goat
442. Car
443. Car
444. Goat
445. Goat
446. Goat
447. Goat
448. Goat
449. Goat
450. Goat
451. Goat
452. Goat
453. Goat
454. Car
455. Goat
456. Goat
457. Car
458. Goat
459. Goat
460. Car
461. Goat
462. Car
463. Goat
464. Goat
465. Car
466. Car
467. Car
468. Goat
469. Car
470. Goat
471. Goat
472. Goat
473. Car
474. Goat
475. Goat
476. Goat
477. Goat
478. Car
479. Car
480. Car
481. Goat
482. Car
483. Car
484. Car
485. Goat
486. Goat
487. Goat
488. Goat
489. Goat
490. Car
491. Goat
492. Goat
493. Goat
494. Goat
495. Goat
496. Goat
497. Goat
498. Car
499. Goat
500. Goat
501. Goat
502. Goat
503. Car
504. Goat
505. Goat
506. Goat
507. Goat
508. Goat
509. Goat
510. Goat
511. Goat
512. Goat
513. Car
514. Goat
515. Car
516. Car
517. Car
518. Car
519. Goat
520. Car
521. Goat
522. Car
523. Goat
524. Goat
525. Goat
526. Goat
527. Goat
528. Goat
529. Car
530. Goat
531. Car
532. Goat
533. Goat
534. Goat
535. Car
536. Car
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Car
545. Car
546. Goat
547. Car
548. Car
549. Goat
550. Car
551. Goat
552. Goat
553. Goat
554. Car
555. Goat
556. Car
557. Goat
558. Goat
559. Car
560. Goat
561. Goat
562. Goat
563. Goat
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Goat
570. Car
571. Car
572. Goat
573. Goat
574. Car
575. Car
576. Goat
577. Goat
578. Goat
579. Goat
580. Goat
581. Car
582. Goat
583. Goat
584. Goat
585. Car
586. Goat
587. Car
588. Goat
589. Goat
590. Goat
591. Goat
592. Goat
593. Car
594. Goat
595. Goat
596. Goat
597. Goat
598. Goat
599. Car
600. Goat
601. Car
602. Goat
603. Goat
604. Car
605. Car
606. Car
607. Car
608. Goat
609. Goat
610. Goat
611. Goat
612. Goat
613. Goat
614. Goat
615. Car
616. Goat
617. Car
618. Goat
619. Car
620. Goat
621. Goat
622. Goat
623. Goat
624. Car
625. Goat
626. Goat
627. Goat
628. Car
629. Car
630. Goat
631. Goat
632. Goat
633. Car
634. Goat
635. Goat
636. Goat
637. Car
638. Goat
639. Goat
640. Car
641. Car
642. Goat
643. Car
644. Goat
645. Goat
646. Car
647. Goat
648. Car
649. Goat
650. Goat
651. Goat
652. Goat
653. Goat
654. Goat
655. Car
656. Car
657. Car
658. Goat
659. Car
660. Goat
661. Car
662. Car
663. Goat
664. Goat
665. Goat
666. Goat
667. Goat
668. Goat
669. Goat
670. Goat
671. Car
672. Goat
673. Goat
674. Goat
675. Car
676. Car
677. Goat
678. Goat
679. Car
680. Goat
681. Car
682. Goat
683. Car
684. Goat
685. Goat
686. Goat
687. Goat
688. Goat
689. Goat
690. Goat
691. Goat
692. Car
693. Goat
694. Goat
695. Car
696. Goat
697. Car
698. Car
699. Car
700. Goat
701. Car
702. Car
703. Car
704. Goat
705. Goat
706. Goat
707. Goat
708. Goat
709. Car
710. Goat
711. Car
712. Goat
713. Goat
714. Goat
715. Goat
716. Goat
717. Goat
718. Car
719. Car
720. Car
721. Goat
722. Goat
723. Goat
724. Goat
725. Car
726. Goat
727. Car
728. Car
729. Goat
730. Goat
731. Goat
732. Goat
733. Goat
734. Car
735. Car
736. Car
737. Goat
738. Goat
739. Car
740. Goat
741. Car
742. Goat
743. Goat
744. Goat
745. Goat
746. Car
747. Goat
748. Goat
749. Car
750. Goat
751. Goat
752. Goat
753. Goat
754. Goat
755. Goat
756. Car
757. Goat
758. Goat
759. Car
760. Goat
761. Car
762. Goat
763. Goat
764. Goat
765. Goat
766. Goat
767. Car
768. Goat
769. Goat
770. Goat
771. Car
772. Goat
773. Car
774. Goat
775. Car
776. Car
777. Car
778. Goat
779. Goat
780. Car
781. Goat
782. Car
783. Goat
784. Goat
785. Goat
786. Goat
787. Goat
788. Car
789. Car
790. Goat
791. Car
792. Goat
793. Car
794. Car
795. Goat
796. Goat
797. Goat
798. Goat
799. Goat
800. Goat
801. Goat
802. Car
803. Goat
804. Car
805. Car
806. Goat
807. Goat
808. Goat
809. Car
810. Car
811. Goat
812. Car
813. Goat
814. Goat
815. Goat
816. Car
817. Goat
818. Goat
819. Goat
820. Goat
821. Goat
822. Goat
823. Car
824. Goat
825. Goat
826. Goat
827. Car
828. Goat
829. Goat
830. Goat
831. Car
832. Goat
833. Car
834. Car
835. Goat
836. Goat
837. Goat
838. Car
839. Goat
840. Goat
841. Goat
842. Car
843. Goat
844. Goat
845. Goat
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Goat
852. Goat
853. Car
854. Goat
855. Goat
856. Car
857. Goat
858. Car
859. Goat
860. Car
861. Car
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Car
868. Goat
869. Car
870. Goat
871. Car
872. Goat
873. Car
874. Goat
875. Goat
876. Goat
877. Goat
878. Goat
879. Goat
880. Goat
881. Car
882. Goat
883. Goat
884. Car
885. Car
886. Goat
887. Goat
888. Goat
889. Car
890. Car
891. Car
892. Goat
893. Goat
894. Car
895. Goat
896. Goat
897. Car
898. Goat
899. Car
900. Goat
901. Goat
902. Car
903. Goat
904. Goat
905. Car
906. Goat
907. Car
908. Goat
909. Goat
910. Car
911. Goat
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Car
919. Goat
920. Goat
921. Car
922. Car
923. Goat
924. Goat
925. Car
926. Goat
927. Goat
928. Goat
929. Car
930. Goat
931. Goat
932. Car
933. Car
934. Goat
935. Car
936. Goat
937. Goat
938. Car
939. Goat
940. Goat
941. Goat
942. Goat
943. Goat
944. Car
945. Car
946. Goat
947. Goat
948. Goat
949. Car
950. Goat
951. Car
952. Goat
953. Goat
954. Goat
955. Car
956. Goat
957. Car
958. Goat
959. Car
960. Goat
961. Goat
962. Goat
963. Goat
964. Goat
965. Goat
966. Car
967. Car
968. Goat
969. Car
970. Goat
971. Car
972. Goat
973. Goat
974. Goat
975. Goat
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Car
982. Goat
983. Goat
984. Goat
985. Goat
986. Car
987. Goat
988. Goat
989. Goat
990. Goat
991. Car
992. Car
993. Car
994. Car
995. Goat
996. Goat
997. Goat
998. Car
999. Car
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 98 times and lost 902 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 40 times and lost 60 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}