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 3 times and lost 7 times.

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

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 343 times and lost 657 times.

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