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. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Car
9. Goat
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 332 times and lost 668 times.

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

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 102 times and lost 898 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 34 times and lost 66 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;
	}
}