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. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Car
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 342 times and lost 658 times.

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